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BELL SYSTEM TECHNICAL JOURNAL 



State and prove this theorem. Suppose that M and iV (Fig. 6) are two 

 (w + 1) terminal networks, M having the hindrance functions Uk (k = 

 1, 2, • • • w) between terminals a and k, and N having the functions Vk 

 between b and k. Further, let M be such that Ujk = l(j') ^ = 1. 2, • • • , n). 

 We will say, in this case, that M is a disjunctive network. Under these con- 

 ditions we shall prove the following: 



Theorem 1: If the corresponding terminals I, 2, • • • , ti of M and N are 

 connected together, then 



Uab=^Il{Uk-\- Vk) 



(1) 



where Uab is the hindrance from terminal a to terminal b. 



Fig. 6 — Network for general design theorem. 



Proof: It is known that the hindrance Uab may be found by taking the 

 product of the hindrances of all possible paths from atob along the elements 

 of the network.* We may divide these paths into those which cross the Hne 

 L once, those which cross it three times, those which cross it five times, etc. 

 Let the product of the hindrances in the first class be TFi , in the second 

 class Ws , etc. Thus 



Now clearly 



Uab= Wi-WrW, 



Wi = n (U + Vk) 



(2) 



and also 



Ws = TFb 



1 



since each term in any of these must contain a summand of the type Ujk 

 which we have assumed to be 1. Substituting in (2) we have the desired 

 result. V, 



The method of using this theorem to synthesize networks may be roughly 



